Question: Given $ \overrightarrow{PQ}\perp\overrightarrow{PS}$, $ m \angle RPS = 4x + 37$, and $ m \angle QPR = 5x + 17$, find $m\angle RPS$. $P$ $Q$ $S$ $R$
Explanation: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Since we are given that $\overrightarrow{PQ}\perp\overrightarrow{PS}$ , we know ${m\angle QPS = 90}$ Substitute in the expressions that were given for each measure: $ {5x + 17} + {4x + 37} = {90}$ Combine like terms: $ 9x + 54 = 90$ Subtract $54$ from both sides: $ 9x = 36$ Divide both sides by $9$ to find $x$ $ x = 4$ Substitute $4$ for $x$ in the expression that was given for $m\angle RPS$ $ m\angle RPS = 4({4}) + 37$ Simplify: $ {m\angle RPS = 16 + 37}$ So ${m\angle RPS = 53}$.